Quantum Theory of Many-Body Systems
Techniques and Applications
Published on 1. April 1998
Book
Hardback
XV, 229 pages
978-0-387-98384-4 (ISBN)
Description
Intended for graduate students in physics and related fields, this text is a self contained treatment of the physics of many-body systems from the point of view of condensed matter. The approach, quite traditionally, uses the mathematical formalism of quasiparticles and Green's functions. In particular, it covers all the important diagram techniques for normal and superconducting systems, including the zero- temperature perturbation theory, and the Matsubara, Keldysh, and Nambu -Gor'kov formalisms. The aim is not to be exhaustive, but to present just enough detail to enable the student to follow the current research literature or to apply the techniques to new problems. Many of the examples are drawn from mesoscopic physics, which deals with systems small enough that quantum coherence is maintained throughout their volume, and which therefore provides an ideal testing ground for many-body theories. The book begins by introducing the Green's function for one-particle systems (using Feynman path integrals), general perturbation theory, and second quantization. It then turns to the usual zero-temperature formalism, discussing the properties and physical meaning of the Green's function for many-body systems and then developing the diagram techniques of perturbation theory. The theory is extended to finite temperatures, including a discussion of the Matsubara formalism as well as the Keldysh technique for essentially nonequilibrium systems. The final chapter is devoted to applications of the techniques to superconductivity, incuding discussions of the superconducting phase transition, elementary excitations, transport, Andreev reflections, and Josephson junctions. Problems at the end of each chapter help to guide learning an to
More details
Series
Edition
1998
Language
English
Place of publication
New York
United States
Target group
College/higher education
Graduate
Product notice
Laminated cover
Illustrations
10 s/w Tabellen
10 black & white tables, biography
Dimensions
Height: 23.5 cm
Width: 15.5 cm
Thickness: 15 mm
Weight
1180 gr
ISBN-13
978-0-387-98384-4 (9780387983844)
DOI
10.1007/978-1-4612-0595-1
Schweitzer Classification
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Content
1 Basic Concepts.- 1.1 Introduction: Whys and Hows of Quantum Many-Body Theory.- 1.1.1 Screening of Coulomb Potential in Metal.- 1.1.2 Time-Dependent Effects. Plasmons.- 1.2 Propagation Function in a One-Body Quantum Theory.- 1.2.1 Propagator: Definition and Properties.- 1.2.2 Feynman's Formulation of Quantum Mechanics: Path (Functional) Integrals.- 1.2.3 Quantum Transport in Mesoscopic Rings: Path Integral Description.- 1.3 Perturbation Theory for the Propagator.- 1.3.1 General Formalism.- 1.3.2 An Example: Potential Scattering.- 1.4 Second Quantization.- 1.4.1 Description of Large Collections of Identical Particles. Fock's Space.- 1.4.2 Bosons.- 1.4.3 Number and Phase Operators and Their Uncertainty Relation.- 1.4.4 Fermions.- 1.5 Problems to Chapter 1.- 2 Green's Functions at Zero Temperature.- 2.1 Green's Function of The Many-Body System: Definition and Properties.- 2.1.1 Definition of Green's Functions of the Many-Body System.- 2.1.2 Analytic Properties of Green's Functions.- 2.1.3 Retarded and Advanced Green's Functions.- 2.1.4 Green's Function and Observables.- 2.2 Perturbation Theory: Feynman Diagrams.- 2.2.1 Derivation of Feynman Rules. Wick's and Cancellation Theorems.- 2.2.2 Operations with Diagrams. Self Energy. Dyson's Equation.- 2.2.3 Renormalization of the Interaction. Polarization Operator.- 2.2.4 Many-Particle Green's Functions. Bethe-Salpeter Equations. Vertex Function.- 2.3 Problems to Chapter 2.- 3 More Green's Functions, Equilibrium and Otherwise, and Their Applications.- 3.1 Analytic Properties of Equilibrium Green's Functions.- 3.1.1 Statistical Operator (Density Matrix). The Liouville Equation.- 3.1.2 Definition and Analytic Properties of Equilibrium Green's Functions.- 3.2 Matsubara formalism.- 3.2.1 Bloch's Equation.- 3.2.2 Temperature (Matsubara) Green's Function.- 3.2.3 Perturbation Series and Diagram Techniques for the Temperature Green's Function.- 3.3 Linear Response Theory.- 3.3.1 Linear Response Theory. Kubo Formulas.- 3.3.2 Fluctuation-Dissipation Theorem.- 3.4 Nonequilibrium Green's Functions.- 3.4.1 Nonequilibrium causal Green's function: definition.- 3.4.2 Contour Ordering and Three More Nonequilibrium Green's Functions.- 3.4.3 The Keldysh Formalism.- 3.5 Quantum Kinetic Equation.- 3.5.1 Dyson's Equations for Nonequilibrium Green's Functions.- 3.5.2 The Quantum Kinetic Equation.- 3.6 Application: Electrical Conductivity of Quantum Point Contacts.- 3.6.1 Quantum Electrical Conductivity in the Elastic Limit.- 3.6.2 Elastic Resistance of a Point Contact: Sharvin Resistance, the Landauer Formula, and Conductance Quantization 133.- 3.6.3 The Electron-Phonon Collision Integral in 3 D Quantum Point Contact.- 3.6.4 *Calculation of the Inelastic Component of the Point Contact Current.- 3.7 Method of Tunneling Hamiltonian.- 3.8 Problems to Chapter 3.- 4 Methods of the Many-Body Theory in Superconductivity.- 4.1 Introduction: General Picture of the Superconducting State.- 4.2 Instability of the Normal State.- 4.3 Pairing (BCS) Hamiltonian.- 4.3.1 Derivation of the BCS Hamiltonian.- 4.3.2 Diagonalization of the BCS Hamiltonian. The Bogoliubov transformation. Bogoliubov-de Gennes Equations..- 4.3.3 Bogolons.- 4.3.4 Thermodynamic Potential of a Superconductor.- 4.4 Green's Functions of a Superconductor: The Nambu-Gor'kov Formalism.- 4.4.1 Matrix Structure of the Theory.- 4.4.2 Elements of the Strong Coupling Theory.- 4.4.3 Gorkov's Equations for the Green's Functions.- 4.4.4 Current-Carrying State of the Superconductor.- 4.4.5 Destruction of Superconductivity by Current.- 4.5 Andreev Reflection.- 4.5.1 The Proximity Effect in a Normal Metal in Contact With a Superconductor.- 4.5.2 Andreev Levels and Josephson Effect in a Clean SNS Junction.- 4.5.3 Josephson Current in a Short Ballistic Junction. Quantization of Critical Current in Quantum Point Contact.- 4.5.4 Josephson Current in a Long SNS Junction.- 4.5.5 *Transport in Superconducting Quantum Point Contact: The Keldysh Formalism Approach..- 4.6 Tunneling of Single Electrons and Cooper Pairs.- 4.6.1 Coulomb Blockade of Single-Electron Tunneling.- 4.6.2 Superconducting Grain: When One Electron Is Too Many..- 4.7 Problems to Chapter 4.- A Landauer Formalism for Hybrid Normal-Superconducting.- Structures.- A.1 The Landauer-Lambert formula.- A.2 Giant Conductance Oscillations in Ballistic Andreev Interferometers.- References.